We have .
Let’s first prove that is symmetric:
So we have by uniqueness of the inverse.
Proving that is a projection matrix:
We look for with and .
Now we have different cases:
First column (j = 1)
So the first column has 1 in the first row, and 0 in all the others.
First row (i = 1)
So the first row has 1 in the first column and in the others.
Diagonal (i = j)
So the diagonal is all 1.
General values (i != j) (i != 1) (j != 1)
We want to analyze . Let’s try to get the elements of more formally. We know that
so we will have
Analyzing the product:
Let’s analyze the cases separately:
i = 1 and j = 1
i = 1 and j != 1
i != 1 and j = 1
i != 1 and j != 1
If we use
then we get
finishing the proof.